Knot theory in 3 -manifold via virtual knot theory Teruhisa Kadokami - - PowerPoint PPT Presentation

Knot theory in 3-manifold via virtual knot theory

Teruhisa Kadokami (Kanazawa University)

Mathematics of Knots II December 20th, 2019 (Fri.) Nihon University, College of Humanities and Sciences

2/32

Contents §0. Introduction

4 – 4

Part I : Geometric virtual knot theory §1. Diagrammatic definition of virtual link

6 – 13

§2. Geometric translation : Kuperberg’s theorem

14 – 22

Part II : Knot theory in 3-manifold §3. Compression body decomposition of compact 3-manifolds

24 – 27

§4. Knot theory in 3-manifold via virtual knot theory

28 – 30

§5. Problems

31 – 31

3/32

References

[1] L. Kauffman Virtual knot theory European Jounal of Combinatorics 20 (1999), 663–690. [2] N. Kamada and S. Kamada Abstract link diagrams and virtual knots

• J. Knot Theory Ramifications 9 (2000), 93–106.

[3] S. Carter, S. Kamada and M. Saito Stable equivalence of knots on surfaces and virtual knot cobordisms

• J. Knot Theory Ramifications 11 (2002), 311–322.

[4] G. Kuperberg What is a virtual link ? Algebraic Geometry & Topology 3 (2003), 587–591. [5] T. Kadokami Classification of closed virtual 2-braids Journal of Knot Theory and its Ramifications 17 (2008), 1223–1239.

4/32

§0. Introduction

• Classical knot theory

link in S3 = ⇒ diagram on S2 = ⇒ invariant Virtual knot theory ? = ⇒ diagram on S2 = ⇒ invariant diagram on S2 = ⇒ diagram on F = ⇒ link in F × [0, 1] = ?

• M : ori. conn. compact 3-manifold

M = V ∪ W : Heegaard splitting F = V ∩ W : Heegaard surface L : link in M = ⇒ L : link in N(F) ∼ = F × [0, 1] ⊂ M

5/32

Part I : Geometric virtual knot theory

6/32

§1. Diagrammatic definition of virtual link Link in S3 (Classical link)

• φ :

n

⨿

i=1

(S1)i → S3 or R3 : embedding = ⇒ L = Im(φ) = K1 ∪ . . . ∪ Kn : n-component link Ki = φ((S1)i) : the i-th component of L

• n = 1 =

⇒ L = K : knot

• ∀(S1)i : oriented =

⇒ L : oriented link

• L, L′ : two links are equivalent (ambient-isotopic) ⇐

⇒

∃F : S3 × [0, 1] → S3 × [0, 1] : level-preserving homeo. s.t.

F0 = idS3 & F1(L) = L′. (i.e. F : ambient-isotopy)

7/32

Projection p : S3 = R3 ∪ {∞} → S2 = R2 ∪ {∞} : projection (1) p((x, y, z)) = (x, y) if (x, y, z) ∈ R3 (2) p(∞) = ∞

c c1 c2

α β

p

p(α) p(β) z O

8/32

Link diagram trefoil 3 figure eight knot 4 Hopf link H 1 2 n trivial knot O trivial link On

1 1

9/32

Reidemeister moves

(R1) (R2) (R3)

10/32

Theorem (Fundamental Theorem of Knot Theory) L, L′ : two links D, D′ : two diagrams of L, L′, respectively L ∼ = L′ : equivalent ⇐ ⇒ D ← → D′ : finite sequence of Reidemeister moves L = {links} ⊃ Ln = {n-component links} D = {link diagrams} ⊃ Dn = {n-component link diagrams} L = D/⟨(R1), (R2), (R3)⟩ ⊃ Ln = Dn/⟨(R1), (R2), (R3)⟩ Φ : D → L, Φn : Dn → Ln : natural projections

11/32

Virtual link diagram real crossing virtual crossing virtual trefoil virtual Hopf link Kishino’s knot

12/32

Virtual Reidemeister moves

(R1) (R2) (R3) (V1) (V2) (V3) (V4)

13/32

V = {virtual links} ⊃ Vn = {n-component virtual links} D = {virtual link diagrams} ⊃ Dn = {n-component virtual link diagrams} V = D/⟨(R1), (R2), (R3), (V1), (V2), (V3), (V4)⟩ ⊃ Vn = Dn/⟨(R1), (R2), (R3), (V1), (V2), (V3), (V4)⟩

14/32

§2. Geometric translation : Kuperberg’s theorem Abstract link diagram [N. Kamada-S. Kamada]

D (N(D), D) ~ ~

15/32

D : virtual link diagram (N( D), D) : the abstract link diagram of D, where N( D) : ori. compact surface canonically obtained from D, and

• D : a diagram on N(

D).

16/32

Surface realization

D (F, D) ~

17/32

D : virtual link diagram (N( D), D) : the abstract link diagram of D (F, D) : a surface realization of D, where F : ori. closed surface obtained from N( D) by attaching compact surfaces to ∂N( D). (F, D) : the canonical realization of D, where F : ori. closed surface obtained from N( D) by attaching disks to ∂N( D).

18/32

Numerical invariants L : virtual link D : diagram of L (F, D) : the canonical realization of D sg(D) = (the sum of genera of components of F) : the supporting genus of D c(D) = (the number of components of F) : the splitting number of D sg(L) = min{sg(D) | D : diagram of L} : the supporting genus of L c(L) = max{c(D) | D : diagram of L} : the splitting number of L

19/32

Minimal realization L : virtual link D : diagram of L (F, D) : a surface realization of D : minimal realization of L ⇐ ⇒ g(F) = sg(L) & (the number of components of F) = c(L). Then D : minimal diagram.

20/32

Space realization L : virtual link D : diagram of L (F, D) : a surface realization of D (F, D) can be regarded as a (framed) link D in F × [0, 1]. (F × [0, 1], D) or (F × [0, 1], L) : space realization of (F, D). If (F, D) : the canonical realization of D = ⇒ (F × [0, 1], D) : space realization of D. If (F, D) : minimal realization of D = ⇒ (F × [0, 1], D) : minimal (space) realization of L.

21/32

Theorem [Kuperberg] L : virtual link (1) (existence)

∃D : minimal diagram of L

(2) (uniqueness) D, D′ : two minimal diagrams of L (F, D), (F ′, D′) : the canonical realizations of D and D′, respectively = ⇒ (F × [0, 1], D), (F ′ × [0, 1], D′) : equivalent links (⇐ ⇒ (F, D), (F ′, D′) are related by an ori.-pres. homeo. φ : F → F ′ & Reidemeister moves on F ′ (i.e. φ( D) and D′ are Reidemeister equivalent on F ′).)

22/32

(H) : attaching a hollow handle to F \ D.

(H)

Theorem We can obtain a minimal realization by a finite sequence

• f (H)−1-moves, and Reidemeister moves on the surface.

Remark L : virtual link, (F × [0, 1], L) : space realization of L φ : F × [0, 1] → (−F) × [1, 0] : natural ori.-pres. homeo. L♯ : virtual link determined from (F × [0, 1], φ( L)) : mixed mirror image of L. Then, in general, L ̸∼ = L♯. To regard (F × [0, 1], L) as a virtual link, we should fix an ori. of F.

23/32

Part II : Knot theory in 3-manifold

24/32

§3. Compression body decomposition of compact 3-manifolds V : compression body ⇐ ⇒ V : tubing (surface)×[0, 1]’s and/or 3-balls by 1-handles ⇐ ⇒ V = (handle body)\ (standard sub-handle bodies) : dual def. M : connected compact oriented 3-manifold M = V ∪ W : ∃compression body decomposition of M F = V ∩ W : Heegaard surface F = ∂+V = ∂+W ∂−V = ∂V \ F, ∂−W = ∂W \ F M = V ∪ W : ordered compression body decomposition of M

25/32

Moves of compression body decompositions (C1) : (stabilization) (C2) : (tubing along a trivial arc) M = V ∪ W − → Σ : a component of ∂−W V ′ : tubing V and N(Σ) along a trivial arc α in W W ′ = M \ V ′ V ∪ W ← → V ′ ∪ W ′ (C3) : (interchanging) V ∪ W ← → W ∪ V Theorem Compression body decompositions of M are related by a finite sequence of (C1), (C2) and (C3).

26/32

(C1)

V V W W

(C2)

V W V W α F F F F

27/32

H = {ordered compression body decomp.s} M = {connected compact oriented 3-manifolds} = H/⟨(C1), (C2), (C3)⟩

• M = H/⟨(C1), (C2)⟩

H

p

− → M

p′

− → M : natural projections H0 = {minimal genus ordered compression body decomp.s} ⊂ H

∀M ∈ M, we take sets

D(M) = (p′ ◦ p)−1(M) ⊃ D0(M) = D(M) ∩ H0 ̸= ∅.

• ex. M = F × [0, 1] =

⇒ D0(M) = {2 points}.

28/32

§4. Knot theory in 3-manifold via virtual knot theory M : connected compact oriented 3-manifold L = K1 ∪ . . . ∪ Kn ⊂ M : link in M M = V ∪ W : ordered compression body decomposition of M F = V ∩ W : Heegaard surface L can be regarded as a link in N(F) ≒ virtual link M = N(F) ∪ (2-handles and 3-handles). “Main Theorem” {links in M} ← → {links in N(F)}/⟨(C1), (C2), (C3), 2-handles⟩

29/32

L ⊂ M − → L ⊂ N(F) − → L as a virtual link : a representing virtual link For L : link in M, s = [M = V ∪ W] ∈ D(M), V(L, s) : the representing virtual links of L in s. V(L, M) = ∪

s∈D(M)

V(L, s), V0(L, M) = ∪

s∈D0(M)

V(L, s), V(M) = ∪

L

V(L, M), V0(M) = ∪

L

V(L, M). s = [V ∪ W]

(C3)

− → s♯ = [W ∪ V ]. Lemma K ∈ V(L, s) = ⇒ K♯ ∈ V(L, s♯). Lemma V(S3) = V(D3) = {virtual links}(= V).

30/32

• V(L, s) = {ℓ ∈ V(L, s) | sg(ℓ) = min{sg(ℓ′) | ℓ′ ∈ V(L, s)}}
• V(L, M) =

∪

s∈D(M)

• V(L, s),
• V0(L, M) =

∪

s∈D0(M)

• V(L, s),
• V(M) =

∪

L

V(L, M),

• V0(M) =

∪

L

V(L, M). Lemma s = [M = V ∪ W] ∈ D(M) & V or W : handlebody = ⇒ V(L, s) consists of classical links.

• ex. (1) M : lens space

T : torus knot in M = ⇒ V0(T, M) = {O}. (2) V0(L, F × [0, 1]) = {1 point}.

31/32

§5. Problems

• Q1. Determine V(L, M), V(M), V0(L, M), V0(M),

V(L, M), V(M),

• V0(L, M),

V0(M).

• Q2. Is

V0(L, M) finite in general ? In particular, ♯ V0(L, M) = 1 in general ?

• Q3. Does

V(L, M)/ V0(L, M) characterize (M, L) ?

• Q4. Does

V(M)/ V0(M) characterize M ?

• Q5. Invariant study.
• Q6. fundamental group −

→ Gordon-Lueke type theorem ?

32/32

Thank you for your attention !